For \(\ell \ge 2\) , joint ergodicity is a much tougher property to establish.
In [1]} it was shown that the collection of sequences \([n^{c_1}],\ldots , [n^{c_\ell }]\) , where \(c_1,\ldots , c_\ell \in {\mathbb {R}}_+\setminus {\mathbb {Z}}\)  are distinct, is jointly ergodic, and further examples involving polynomials with real coefficients and other sequences arising from smooth functions can be found in [2]}, [3]}, [4]}, [5]}. Moreover, a collection of polynomial sequences \(p_1,\ldots , p_\ell \in {\mathbb {Z}}[t]\)  is known to be jointly ergodic
for all totally ergodic systems if and only if the polynomials are rationally independent [6]}; a typical case is the pair \(n,n^2\) , which was first handled in [7]}.
The proofs of most of these results rely on deep tools from ergodic theory, such as the Host-Kra theory of characteristic factors [8]}
and equidistibution results on nilmanifolds. But a general criterion for establishing joint ergodicity, like the one mentioned for \(\ell =1\) , is still lacking, the reason being that we do not have a good substitute for the spectral theorem for unitary operators that would allow us to represent correlation sequences of the form \(\int f_0 \cdot T^{n_1}f_1\cdot \ldots \cdot T^{n_\ell }f_\ell \, d\mu \) , for sparse values of \(n_1,\ldots , n_\ell \in {\mathbb {N}}\) , in a useful way.
